The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.

翻译：熵产生率是非平衡统计物理学中的一个核心量，用于衡量随机过程偏离时间可逆性的程度。本文计算了在扩散过程的时间反转仍保持扩散性质条件下，非平衡稳态扩散过程的熵产生。我们首先刻画了离散时间和连续时间马尔可夫过程的熵产生。我们研究了时间齐次平稳扩散的时间反转，并回顾了扩散性质可逆性的最一般条件，这包括亚椭圆和退化扩散，以及局部利普希茨向量场。我们将漂移分解为时间可逆和不可逆部分，或者等价地，将生成元分解为对称和反对称算子。我们展示了这与超柯西理论中考虑的后向柯尔莫哥洛夫方程分解，以及GENERIC形式的福克-普朗克方程分解的等价性。主要结果表明，当漂移的时间不可逆部分几乎处处位于波动率矩阵的值域内时，过程的前向和反转路径空间测度相互等价，并给出了熵产生的表达式。当这一条件不成立时，测度相互奇异，且熵产生为无穷大。我们通过线性扩散的精确数值模拟验证了这些结果。通过几个例子，我们说明了非线性扩散的熵产生与其数值模拟之间的差异，并阐述了熵产生如何用于精确数值模拟。最后，我们讨论了时间不可逆性与采样效率之间的关系，以及如何修改熵产生的定义来度量过程偏离广义可逆性的程度。